Problem 23 - Beginning and Intermediate Algebra

Question

Find the general term of each geometric sequence. $$5,10,20,40, \dots$$

Step-by-Step Solution

Answer

The general term of the given geometric sequence is \(a_n = 5 \times 2^{n-1}\).

Step 1: Identify the common ratio

To find the common ratio, divide the second term by the first term or any two consecutive terms. In our example, we have: $$\frac{10}{5} = 2$$ So the common ratio (r) is 2.

Step 2: Apply the formula for the general term of a geometric sequence

The general term of a geometric sequence is given by the formula: $$a_n = a_1 \times r^{n-1}$$ where \(a_n\) is the general term, \(a_1\) is the first term, r is the common ratio, and n is the term number.

Step 3: Apply the formula to the given sequence

In our sequence, the first term \(a_1\) is 5 and the common ratio r is 2. Using the formula from Step 2, we can find the general term of the sequence: $$a_n = 5 \times 2^{n-1}$$ Therefore, the general term of the given geometric sequence is \(a_n = 5 \times 2^{n-1}\).

Key Concepts

Understanding the Common Ratio in a Geometric Sequence

Every geometric sequence has what's called a "common ratio." This is a key factor that helps define the sequence itself. To find the common ratio, you simply divide any term in the sequence by its preceding term.
For example, in the sequence given: 5, 10, 20, 40, you take the second term (10) and divide it by the first term (5).
This calculation looks like this: \( \frac{10}{5} = 2 \). Thus, the common ratio here is 2.

• This common ratio tells us how each term is related to its previous one.
• It shows consistent multiplicative growth from one term to the next.
• For example, starting at 5, multiply by 2 to get 10, by 2 again to get 20, and so on.

Understanding this "common factor" is essential for solving various sequence problems. It helps you to predict the pattern quickly and accurately.

Using the General Term Formula to Define Sequences

The general term formula for a geometric sequence is another important concept. This formula allows you to find any term within the sequence without listing all preceding terms.
The formula is expressed as:\[ a_n = a_1 \times r^{n-1} \]Let's break this down:

• \(a_n\) represents the general term or the term you are trying to find.
• \(a_1\) is the first term in the sequence. For our specific sequence, it is 5.
• \(r\) is the common ratio, which we've calculated as 2.
• \(n\) is the position of the term within the sequence.

By applying this formula to our sequence:

• To find the 3rd term, substitute 3 for \(n\): \(a_3 = 5 \times 2^{3-1} = 20\).
• This formula helps you see the exponential growth pattern in geometric sequences.

Solving Sequence Problems Effectively

Sequence problems often arise when trying to determine unknown terms or verify patterns within numeric sequences. Understanding the key concepts behind them, such as the common ratio and the general term formula, is crucial to solving them.
Here’s how you can approach sequence problems:

• First, identify if the sequence is geometric by checking for a constant ratio.
• Calculate the common ratio if any two consecutive terms are given.
• Use the general term formula to find any specific term required by the problem.
• Understanding these elements helps in building functions and models in algebra, calculus, and beyond.

This approach will not only help you solve problems on tests but also apply mathematical reasoning to real-world scenarios, like calculating investments, predicting exponential growth, and more.